Senary

In mathematics, a senary numeral system is a base-6 numeral system.

Senary may be considered useful in the study of prime numbers since all primes other than 2 and 3, when expressed in base-six, have 1 or 5 as the final digit. Writing out the prime numbers in base-six (and using the subscript 6 to denote that these are senary numbers), the first few primes are

$2_{6},3_{6},5_{6},11_{6},15_{6},21_{6},25_{6},31_{6},35_{6},45_{6},51_{6},$

$101_{6},105_{6},111_{6},115_{6},125_{6},\ldots$

That is, for every prime number p with $p\neq 2,3$, one has the modular arithmetic relations that either $p\mod 6=1$ or $p\mod 6=5$: the final digits is a 1 or a 5. This is proved by contradiction. For integer n:

• If n mod 6 = 0, 6|n
• If n mod 6 = 2, 2|n
• If n mod 6 = 3, 3|n
• If n mod 6 = 4, 2|n

Furthermore, all known perfect numbers besides 6 itself have 44 as the final two digits when expressed in base 6.

Finger counting

Main article: Finger counting

Senary multiplication table

*	1	2	3	4	5	10	11	12
1	1	2	3	4	5	10	11	12
2	2	4	10	12	14	20	22	24
3	3	10	13	20	23	30	33	40
4	4	12	20	24	32	40	44	52
5	5	14	23	32	41	50	55	104
10	10	20	30	40	50	100	110	120
11	11	22	33	44	55	110	121	132
12	12	24	40	52	104	120	132	144

Each regular human hand may be said to have six unambiguous positions; a fist, one finger (or thumb) extended, two, three, four and then all five extended.

If the right hand is used to represent a unit, and the left to represent the 'sixes', it becomes possible for one person to represent the values from zero to 55_senary (35_decimal) with their fingers, rather than the usual ten obtained in standard finger counting. e.g. if three fingers are extended on the left hand and four on the right, 34_senary is represented. This is equivalent to 3 × 6 + 4 which is 22_decimal.

Which hand is used for the 'sixes' and which the units is down to preference on the part of the counter, however when viewed from the counter's perspective, using the left hand as the most significant digit correlates with the written representation of the same senary number.

Other finger counting systems, such as chisanbop or finger binary, allow counting to 99, 1,023, or even higher depending on the method (though not necessarily senary in nature). The English monk and historian Bede in the first chapter of De temporum ratione, (725), entitled "Tractatus de computo, vel loquela per gestum digitorum", which allowed counting up to 9,999 on two hands.

Fractions

Because six is the product of the first two prime numbers and is adjacent to the next two prime numbers, many senary fractions have simple representations:

Decimal base Prime factors of the base: 2, 5			Senary base Prime factors of the base: 2, 3
Fraction	Prime factors of the denominator	Positional representation	Positional representation	Prime factors of the denominator	Fraction
1/2	2	0.5	0.3	2	1/2
1/3	3	0.3333... = 0.3	0.2	3	1/3
1/4	2	0.25	0.13	2	1/4
1/5	5	0.2	0.1111... = 0.1	5	1/5
1/6	2, 3	0.16	0.1	2, 3	1/10
1/7	7	0.142857	0.05	11	1/11
1/8	2	0.125	0.043	2	1/12
1/9	3	0.1	0.04	3	1/13
1/10	2, 5	0.1	0.03	2, 5	1/14
1/11	11	0.09	0.0313452421	15	1/15
1/12	2, 3	0.083	0.03	2, 3	1/20
1/13	13	0.076923	0.024340531215	21	1/21
1/14	2, 7	0.0714285	0.023	2, 11	1/22
1/15	3, 5	0.06	0.02	3, 5	1/23
1/16	2	0.0625	0.0213	2	1/24
1/17	17	0.0588235294117647	0.0204122453514331	25	1/25
1/18	2, 3	0.05	0.02	2, 3	1/30

Natural languages

The Ndom language of Papua New Guinea is reported to have senary numerals. Mer means 6, mer an thef means 6×2 = 12, nif means 36, and nif thef means 36×2 = 72. Proto-Uralic is also suspected to have used senary numerals.

See also

• Diceware has a way of encoding base 6 values into pronounceable words, using a standardized list of 7,776 distinct words

Closely related number systems

• Duodecimal (base 12)
• Trigesimal (Base 30)
• Base 36

External links

• Shack's Base Six Dialectic
• Senary Base Conversion, includes fractional part, from Math Is Fun
• MathService, ConvertBase .NET web service
• Base6Clock, A digital base 6 clock